Physics 220: Exercise on non-conservative forces

An object slides on a surface with coefficient of friction m k. If it starts with speed vo, and slides up a slope making angle q with the horizontal, how high does it go?

 

We start with a diagram:

Then we look at the energy of the system. We must consider gravitational potential energy and kinetic energy. Let's put the reference level at the bottom of the slope.

 

Before (bottom of slope)

After (max height)

Grav PE.

0

mgh

Kinetic energy

mv2/2

0

Total mechanical energy

mv2/2

mgh

Now work done by non-conservative forces changes the TME:

Wnon-cons = Ef -EI

The only non-conservative force is friction, and the work done is:

since the frictional force and the displacement are in opposite directions.

Thus:

mgh- mv2/2 = -fs

To proceed, we need f, which we can get from the friction relation f = m kn and the free-body diagram:

f = m kmgcosq

Finally, we use the geometry of the slope to relate h and s. Thus:

mgh- mv2/2 = -fs = - m kmgcosq h/sinq

The result reduces to v2/2g when the coeffcient of friction is zero, or the angle is p /2, as expected. We also get zero when the angle is zero, which is also expected.